Plot the Einstein specific heat of a solid as a function of...

Plot the Einstein specific heat of a solid as a function of temperature and thus reproduce Figure $15.17$. Assume that the vibrational level spacing has the value $\varepsilon=0.037 \mathrm{eV}$ (a typical value in a solid). (b) Using the value $\varepsilon=0.037 \mathrm{eV}$, at what temperature is the heat capacity of the solid given by $0.9$ times the Dulong-Petit value?

Plot the Einstein specific heat of a solid as a function of temperature and thus reproduce Figure $15.17$. Assume that the vibrational level spacing has the value $\varepsilon=0.037 \mathrm{eV}$ (a typical value in a solid).
(b) Using the value $\varepsilon=0.037 \mathrm{eV}$, at what temperature is the heat capacity of the solid given by $0.9$ times the Dulong-Petit value?

Key Concepts

Einstein Model of Specific Heat

The Einstein model treats atoms in a solid as independent quantum harmonic oscillators that all vibrate at the same frequency. This model introduces a quantized energy spectrum with a uniform spacing (denoted as ? in the problem) and provides a framework for calculating the specific heat of a solid as a function of temperature. It explains the deviation of the specific heat from classical predictions at low temperatures as fewer vibrational states are thermally accessible.

Dulong-Petit Law

The Dulong-Petit law is an empirical observation that, at sufficiently high temperatures, the molar specific heat of many solids is approximately constant, typically around 3R (or 3kB per atom). This classical result is recovered in the Einstein model when the thermal energy is much greater than the vibrational quantum of energy (kBT >> ?), leading to full excitation of all vibrational modes.

Vibrational Quantum Energy Levels

In the context of the Einstein model, each atom's vibration is quantized, meaning that the atoms can only possess specific discrete energy values separated by a quantum ?. This quantization becomes significant at temperatures where kBT is comparable to or smaller than ?, resulting in a reduced number of accessible energy levels and thus a deviation from the classical specific heat value.

Temperature Dependence of Specific Heat

The specific heat in the Einstein model increases with temperature as more vibrational states become thermally accessible, eventually approaching the constant Dulong-Petit limit at high temperatures. At low temperatures, the specific heat drops exponentially due to the limited number of excited vibrational states, which is a direct consequence of the quantized vibration energy levels.

Characteristic Temperature (Einstein Temperature)

The characteristic temperature in the Einstein model, often defined by ?/kB, sets the scale at which quantum effects become significant. When the temperature of the solid is comparable to this characteristic temperature, quantum effects suppress specific heat below its classical value. Solving problems such as finding the temperature at which the specific heat reaches a certain fraction of the Dulong-Petit value often involves determining how kBT compares to the energy spacing ?.

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